Question 156298

# 5


Note: ln(x) also looks like LN(x) (to pronounce it, simply read off the letters "L" "N")


This is the natural log of x. So it is a logarithmic function.


Let's find the y value when {{{x=1}}}  



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(1)}}} Plug in {{{x=1}}}.



{{{y=0}}} Take the natural log of 1 to get 0



So when {{{x=1}}}, {{{y=0}}}. 



----------------------------



Let's find the y value when {{{x=2}}}  



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(2)}}} Plug in {{{x=2}}}.



{{{y=0.693}}} Take the natural log of 2 to get 0.693



So when {{{x=2}}}, {{{y=0.693}}}.




----------------------------



Let's find the y value when {{{x=4}}}  



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(4)}}} Plug in {{{x=4}}}.



{{{y=1.386}}} Take the natural log of 4 to get 1.386



So when {{{x=4}}}, {{{y=1.386}}}. 


-------------------------------


Let's find the y value when {{{x=8}}} 



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(8)}}} Plug in {{{x=8}}}.



{{{y=2.079}}} Take the natural log of 8 to get 2.079



So when {{{x=8}}}, {{{y=2.079}}}. 



-------------------------------------



Let's find the y value when {{{x=16}}}  



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(16)}}} Plug in {{{x=16}}}.



{{{y=2.773}}} Take the natural log of 16 to get 2.773



So when {{{x=16}}}, {{{y=2.773}}}.



Now let's make a table of the values we just found.




<h4>Table of Values:</h4><pre>

<TABLE border="1" width="100">
<TR><TD>x</TD><TD>y</TD></TR><tr><td>1</td><td>0</td></tr>
<tr><td>2</td><td>0.693</td></tr>
<tr><td>4</td><td>1.386</td></tr>
<tr><td>8</td><td>2.079</td></tr>
<tr><td>16</td><td>2.773</td></tr>
</TABLE>

</pre>


Since the natural log function is logarithmic, this means that the growth is logarithmic. This growth rate is slower than linear growth rate and is the slowest growth rate than all of the growth rates.


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So here's the order of function's growth from smallest growth rate to largest growth rate


{{{y=ln(x)}}}, {{{y=5x-3}}}, {{{y=x^2-3x+2}}}, and {{{y=2x^3+7x^2-x-1}}}, and {{{y=10^x}}}